Solving for a RV as a function of iid RVs

**Reploid** · 09-07-2012 08:34 PM

Hi folks,

I'm not a probability theory guy (economics PhD student), but I've recently gotten interested in networks and I've set up a model that I have no idea how to solve. I'm trying to maximize $E[a^{O_i}]$ for a scalar $a$ and RV $O_i$ which depends on $a$ . In order to find first order conditions, I'm trying to solve for $O_i$ satisfying the following:

$F_{O_0} = F_Y$

Where

$Y=\sum_{i = 1}^{\mu(2n)+1}(O_i * a_i * b_i)+c$
$O_i$ are iid $\forall i$ .
$a_i$ and $b_i$ are iid Bernoulli random variables with $E[a_i]=p$ and $E[b_i]=q$ $\forall i$ .
$c$ is also a Bernoulli random variable.
$\mu$ is a random variable distributed Poisson.
All random variables are independent.
$F_X$ is the CDF of $X$ .
$1=a+n$

I'm not really sure how to approach solving this. If you know a method to solve this, a good reference, or that it's simply not possible, I'd be grateful to hear your feedback. I'd like to find an exact solution, if possible.

Thanks for your consideration.

**chiro** · 09-15-2012 03:08 AM

Hey Reploid.

The first thing I need to ask is if O_i's are discrete random variables and regardless of this answer, what is the domain of the O_i random variables.

If all are discrete, you can get a probability generating function for your Y (even if you don't know the probabilities of O where you represent them symbolically) and then you can impose as many constraints as required to get the actual values of these probabilities.

http://en.wikipedia.org/wiki/Probabi...ating_function

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